Condon domain phase diagram for silver
We present the Condon domain phase diagram for a silver single crystal measured in magnetic fields up to 28 T and temperatures down to 1.3 K. A standard ac method with a pickup coil system is used at low frequency for the measurements of the de Haas–van Alphen effect (dHvA). The transition point fro...
Saved in:
| Date: | 2011 |
|---|---|
| Main Authors: | , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2011
|
| Series: | Физика низких температур |
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/118437 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Condon domain phase diagram for silver / R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, W. Joss // Физика низких температур. — 2011. — Т. 37, № 1. — С. 50–55. — Бібліогр.: 18 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-118437 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1184372025-02-09T16:41:31Z Condon domain phase diagram for silver Kramer, R.B.G. Egorov, V.S. Gasparov, V.A. Jansen, A.G.M. Joss, W. Низкотемпературный магнетизм We present the Condon domain phase diagram for a silver single crystal measured in magnetic fields up to 28 T and temperatures down to 1.3 K. A standard ac method with a pickup coil system is used at low frequency for the measurements of the de Haas–van Alphen effect (dHvA). The transition point from the state of homogeneous magnetization to the inhomogeneous Condon domain state (CDS) is found as the point where a small irreversibility in the dHvA magnetization arises, as manifested by an extremely nonlinear response in the pickup voltage showing threshold character. The third harmonic content in the ac response is used to determine with high precision the CDS phase boundary. The experimentally determined Condon domain phase diagram is in good agreement with the theoretical prediction calculated by the standard Lifshitz–Kosevich formula. We are grateful to I. Sheikin and V.P. Mineev for fruitful discussions. 2011 Article Condon domain phase diagram for silver / R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, W. Joss // Физика низких температур. — 2011. — Т. 37, № 1. — С. 50–55. — Бібліогр.: 18 назв. — англ. 0132-6414 PACS: 75.45.+j, 71.70.Di, 75.60.–d https://nasplib.isofts.kiev.ua/handle/123456789/118437 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Низкотемпературный магнетизм Низкотемпературный магнетизм |
| spellingShingle |
Низкотемпературный магнетизм Низкотемпературный магнетизм Kramer, R.B.G. Egorov, V.S. Gasparov, V.A. Jansen, A.G.M. Joss, W. Condon domain phase diagram for silver Физика низких температур |
| description |
We present the Condon domain phase diagram for a silver single crystal measured in magnetic fields up to 28 T and temperatures down to 1.3 K. A standard ac method with a pickup coil system is used at low frequency for the measurements of the de Haas–van Alphen effect (dHvA). The transition point from the state of homogeneous magnetization to the inhomogeneous Condon domain state (CDS) is found as the point where a small irreversibility in the dHvA magnetization arises, as manifested by an extremely nonlinear response in the pickup voltage showing threshold character. The third harmonic content in the ac response is used to determine with high precision the CDS phase boundary. The experimentally determined Condon domain phase diagram is in good agreement with the theoretical prediction calculated by the standard Lifshitz–Kosevich formula. |
| format |
Article |
| author |
Kramer, R.B.G. Egorov, V.S. Gasparov, V.A. Jansen, A.G.M. Joss, W. |
| author_facet |
Kramer, R.B.G. Egorov, V.S. Gasparov, V.A. Jansen, A.G.M. Joss, W. |
| author_sort |
Kramer, R.B.G. |
| title |
Condon domain phase diagram for silver |
| title_short |
Condon domain phase diagram for silver |
| title_full |
Condon domain phase diagram for silver |
| title_fullStr |
Condon domain phase diagram for silver |
| title_full_unstemmed |
Condon domain phase diagram for silver |
| title_sort |
condon domain phase diagram for silver |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2011 |
| topic_facet |
Низкотемпературный магнетизм |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118437 |
| citation_txt |
Condon domain phase diagram for silver / R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, W. Joss // Физика низких температур. — 2011. — Т. 37, № 1. — С. 50–55. — Бібліогр.: 18 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kramerrbg condondomainphasediagramforsilver AT egorovvs condondomainphasediagramforsilver AT gasparovva condondomainphasediagramforsilver AT jansenagm condondomainphasediagramforsilver AT jossw condondomainphasediagramforsilver |
| first_indexed |
2025-11-28T01:10:46Z |
| last_indexed |
2025-11-28T01:10:46Z |
| _version_ |
1849994512038887424 |
| fulltext |
© R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, and W. Joss, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 1, p. 50–55
Condon domain phase diagram for silver
R.B.G. Kramer1,2,3, V.S. Egorov1,2,4, V.A. Gasparov5, A.G.M. Jansen6, and W. Joss1,2,7
1LNCMI, CNRS, BP 166, 38042 Grenoble Cedex 9, France
E-mail: roman.kramer@grenoble.cnrs.fr
2Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany
3Institut Néel, CNRS, Université Joseph Fourier, BP 166, 38042 Grenoble Cedex 9, France
4Russian Research Center «Kurchatov Institute», 123182 Moscow, Russia
5Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia
6Service de Physique Statistique, Magnétisme, et Supraconductivité, INAC, CEA, 38054 Grenoble Cedex 9, France
7Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
Received March 31, 2010
We present the Condon domain phase diagram for a silver single crystal measured in magnetic fields up to 28 T
and temperatures down to 1.3 K. A standard ac method with a pickup coil system is used at low frequency for
the measurements of the de Haas–van Alphen effect (dHvA). The transition point from the state of homogeneous
magnetization to the inhomogeneous Condon domain state (CDS) is found as the point where a small irreversi-
bility in the dHvA magnetization arises, as manifested by an extremely nonlinear response in the pickup voltage
showing threshold character. The third harmonic content in the ac response is used to determine with high preci-
sion the CDS phase boundary. The experimentally determined Condon domain phase diagram is in good agree-
ment with the theoretical prediction calculated by the standard Lifshitz–Kosevich formula.
PACS: 75.45.+j Macroscopic quantum phenomena in magnetic systems;
71.70.Di Landau levels;
75.60.–d Domain effects, magnetization curves, and hysteresis.
Keywords: domain phase diagram, de Haas–van Alphen effect, Condon domain state.
1. Introduction
The formation of dia- and paramagnetic domains has
been predicted by Condon [1] to occur in nonmagnetic pure
metals by considering the collective interaction between the
electrons on Landau-quantized energy levels in the de Haas–
van Alphen (dHvA) effect. These domains corresponding to
an inhomogeneous magnetization are commonly called
Condon domains. The domain formation results from a self-
consistent treatment of the oscillating dHvA magnetization
M due to the orbital quantization of the electronic system
in the total magnetic induction 0= ( )B H Mμ + , where
= ( )M M B depends on the total induction B in an applied
magnetic field H. Following the Pippard–Shoenberg concept
of magnetic interaction [2,3] where the electrons experience
the influence of the magnetic field induced magnetization of
all neighboring electrons, a thermodynamic instability arises
when the dHvA amplitude becomes large enough, i.e., the
differential susceptibility
0= > 1.M
B
∂
χ μ
∂
(1)
This condition corresponds to the situation where the am-
plitude of the magnetization amplitude becomes compara-
ble to the magnetic field period of the dHvA effect. For
sufficiently strong magnetization amplitudes, this instabili-
ty condition, rewritten like 0 / = 1 < 0H Bμ ∂ ∂ −χ , occurs
in a certain field interval within the paramagnetic part of
each dHvA cycle. In these field intervals, where
0 / < 0H Bμ ∂ ∂ , the induction as function of the applied
field ( )B H is multi-valued, like the van der Waals iso-
therm for a real gas. The system avoids this instability in
the same way as the real gas. For an infinite long rod-like
sample (demagnetization factor = 0n ), the induction B
undergoes a discontinuous transition between the two sta-
ble states with the induction 1B and 2B at a certain ap-
plied field ,H like the liquid–gas specific volumes change
discontinuously at the equilibrium vapor pressure. Both
Condon domain phase diagram for silver
Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 51
stable states 1B and 2B correspond to the same free ener-
gy and the inductions in the instability interval ( 1 2,B B ) are
never realized. Figure 1,a shows schematically the magne-
tization energy
2
mag 0
0
1= ( )
2
E B H−μ
μ
for = 0n and the oscillating dHvA energy described by
the Lifshitz–Kosevich (LK) formula osc = cos(2 / )E a F Bπ
to its simplest approximation with a the oscillation ampli-
tude and F the dHvA frequency. The sum of both ener-
gies as function of B is shown for three different magnetic
fields in Figs. 1,b–d. Usually, there is only one minimum
in the total energy for a given applied magnetic field and
the system will assume this value of B. However if the
curvature of the magE is smaller than the curvature of
oscE two minima coexist at an applied field 1H (Fig. 1,c)
and the induction will jump discontinuously from the value
1B to 2B when sweeping the magnetic field through 1H .
For a plate-like sample, oriented normally to the applied
magnetic field H ( = 1n ), the boundary condition
0 0= [ (1 ) ] =B H n M Hμ + − μ is required even in the inter-
val 1 0 2< <B H Bμ . Therefore, the induction B cannot
change discontinuously and a homogeneous state is not
longer possible. The plate breaks up into domains of oppo-
site magnetization. The volume fractions of the domains
with the respective inductions 1B and 2B are adjusted in a
way that for the average induction of the sample 0=B Hμ
is fulfilled [1]. The regions with 1 0<B Hμ are diamagnet-
ic, those with 2 0>B Hμ are paramagnetic. The domain
walls between the phases 1B and 2B run parallel to H
across the plate. In contrast to magnetic domains in com-
mon ferro- and antiferromagnetism, Condon domains do
not have their origin in the interaction of electrons via their
spin moments, but via their orbital motion.
For a sample shape with intermediate demagnetization
factor, 0 < < 1,n the above-mentioned interval of magnet-
ic field 1 0 2< <B H Bμ with the occurrence of the instabil-
ity will be reduced compared to the plate-like sample with
the field range of this interval proportional to n. Therefore,
samples of arbitrary shape will still show the nonuniform
domain state with the same dia- and paramagnetic phases,
1B and 2 ,B whose domain structure might however be
more complex.
Besides the analogy with the van der Waals gas, there is
a close analogy of the CDS with the intermediate state of
type-I superconductors, where the same boundary condi-
tion of a magnetic field applied to a sample of nonzero
demagnetization factor leads to the formation of alternating
domains in the normal and superconducting state. Condon
domains, however, have the unique feature that the transi-
tion between the uniform and the inhomogeneous domain
state occurs periodically in subsequent dHvA oscillations.
Equation (1) defines the boundary between the uniform
and the Condon domain state. The resulting CDS phase
diagram in the ( , )H T plane can be predicted by means of
the LK formula for the oscillatory magnetization of the
dHvA signal using the Fermi surface parameters, like the
curvature 2 2= /A A k′′ ∂ ∂ of the Fermi surface cross-
section A and the effective mass *,m and the Dingle tem-
perature DT as a parameter for the impurity-scattering
damping of the signal [4].
Up to now, Condon domains have been observed by
different experimental methods: by NMR [5], muon spin
rotation (μSR) spectroscopy [6,7] and, more recently, by
local Hall probes [8]. All experimental observations have
in common that two distinct inductions 1B and 2B or an
induction splitting 2 1=B B Bδ − are measured at a given
applied field H and temperature .T However, these mea-
surements yielded only a few points well inside the ( , )H T
diagram where Condon domains exist that could be com-
pared with the theoretically predicted diagram. As a conse-
quence, for example, the data on beryllium obtained by
μSR required new phase diagram calculations with a mod-
ified LK formula for the susceptibility [9]. The exact deter-
mination of the CDS phase boundary, where Bδ approaches
zero, is difficult and time-consuming with a difference mea-
surement of 1B and 2B [10].
It was shown recently that a small hysteresis occurs in
the measured dHvA signal upon passing the CDS phase
boundary [11]. Due to the irreversible magnetization, an
extremely nonlinear response to a small modulation field
arises in standard ac susceptibility measurements. The
Fig. 1. (Color online) (a) Schematic representation of the magne-
tization energy magE and the dHvA energy oscE as the function
of B for = 0.n (b–d) Sum of these energies for three applied
magnetic fields 0 1 2< < .H H H If the curvature of the parabola
is smaller than the curvature of the oscillating energy two minima
coexist at 1.H This leads to discontinuous jump of the induction
when sweeping the magnetic field through 1.H
E
Emag
Eosc
E Emag osc+E
B
�0 0H B �0 0H B
a b
c d
�0 1H B �0 2H B
E Emag osc+
E
E Emag osc+
E
BB1 B2
R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, and W. Joss
52 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1
Fig. 2. (Color online) Schematic representation of two minor
hysteresis loops. The ac response upon field modulation is win-
dow-like and slightly shifted in phase. The normalized response
(apparent susceptibility) decreases sharply when the modulation
amplitude becomes of the order of the hysteresis loop width.
H
M
out-of-phase signal and the third harmonic of the pickup
voltage rise steeply at the transition point to the CDS. The
threshold character of these quantities offers therefore a
possibility to measure a Condon domain phase diagram.
One should note that the third harmonic of the susceptibili-
ty is commonly used as a very sensitive tool to detect
phase boundaries also of other systems like, e.g., the vor-
tex-glass transition in superconductors [12].
In this article we determine the Condon domain phase
diagram for silver using the third harmonic of the ac sus-
ceptibility for the detection of the nonlinear magnetic re-
sponse. It was shown earlier [13] that detailed calculations
of the magnetoquantum oscillations in silver based on the
LK formula are in good agreement with experimental
dHvA data up to 10 T. This is certainly due to the nearly
spherical Fermi surface of silver. Expecting a good agree-
ment with the theoretically determined CDS phase dia-
gram, we applied to silver this first detailed determination
of the phase diagram.
2. Experimental
The measurements were performed on a high quality sil-
ver single crystal of 4.1×2.1×1.0 mm. The sample was cut
from the same piece than the sample used for the direct ob-
servation of Condon domains using local Hall probe detec-
tion [8]. The sample preparation is described in detail else-
where [14,15]. The sample has a residual resistance ratio
300 K 4.2 K/ =R R 1.6⋅104, measured by the contactless Zer-
nov–Sharvin method [16]. The high quality of the sample
results in a very low Dingle temperature, which was esti-
mated from standard dHvA analysis to be about =DT 0.2 K
yielding an electronic mean free path of about 0.8 mm.
A standard ac modulation method with a compensated
pickup coil system was used. Both pickup coils are iden-
tical and consist of about 400 turns. A long coil wound by
a copper wire produced the modulation field with variable
amplitude at frequencies of 20–200 Hz. The pickup voltage
was simultaneously measured by two lock-in amplifiers on
the first and on higher harmonics. The measurements were
performed in a superconducting coil up to 16 T as well as
in a resistive coil up to 28 T at temperatures of 1.3–4.2 K.
The long side of the sample was parallel to the [100] axis of
the single crystal and was slightly tilted (~5 deg) with re-
spect to the direction of the applied magnetic field so that
only the dHvA frequency from the «belly» orbit of 47300 T
existed in the frequency spectrum.
The method of nonlinear detection, we use here, is ap-
plied to determine for the first time a CDS phase diagram
over a broad range of temperatures and magnetic fields.
Therefore, we will present carefully the technical details of
the measurements in order to show the robustness of the
phase diagram determination with respect to changing ex-
perimental parameters and measurement conditions.
3. Evidence of hysteresis in silver
The employed method to determine a Condon domain
phase diagram is based on the appearance of hysteresis in
the CDS which was first discovered on beryllium [11].
Hysteresis appears at the phase transition to the CDS and
this results in some radical changes in the response to an ac
modulation field. In the following, we will show that the
characteristic nonlinear features in the ac response, as
found in beryllium, are also observed in silver.
In presence of hysteresis the amplitude of the suscepti-
bility, normalized on the modulation level, depends on the
modulation amplitude. This is expected to occur when the
modulation level is of the order of the hysteresis loop
width. The schematic representation of a hysteresis loop in
Fig. 2 explains this nonlinear response to an ac field mod-
ulation. As a result, after the transition to the CDS, the
positive (paramagnetic) part of the susceptibility turns out
to be reduced. From a comparison of two normalized sus-
ceptibilities, one measured with high and the other with
low modulation level, we can in principle find where the
amplitude reduction starts and thereby the transition point
to the CDS.
Figure 2 shows as well that the response to a sinusoidal
field modulation becomes window shaped and is slightly
shifted in phase with respect to the input. Therefore, both
the third harmonic and the out-of-phase signal of the
pickup voltage increase steeply when the CDS phase
boundary is crossed [11]. This threshold behavior offers a
simple way to determine the transition point of the CDS.
The major advantage of third harmonic and out-of-phase
part measurements is that only one magnetic field or tem-
perature sweep through the transition is needed.
Figures 3 and 4 show the above discussed nonlinear fea-
tures in the pickup signal at constant temperature T = 2.7 K
Condon domain phase diagram for silver
Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 53
Fig. 4. (Color online) (a) Phase angle of the third harmonic show-
ing clearly the transition between noise outside the CDS and a
fixed phase in the CDS. (b) The out-of-phase part of the first
harmonic response changes due to the arising hysteresis. Both
data measured at 2.7 K and 0.2 G modulation level.
180
0
–180
0
–0.4
8 141210
�0H, T
an
g
le
3
h
ar
m
.
rd
V
p
,d
eg
Im
,
V
V
P
�
a
b
measured in the superconducting coil. Figure 3,a shows
two traces of the normalized pickup voltage, i.e., the suscep-
tibility, obtained in the same conditions with 1.0 and 0.2 G
modulation amplitude at 160 Hz modulation frequency. In
principle, both modulation levels are small enough com-
pared to the dHvA period of about 20 G at 10 T that iden-
tical traces are expected for the susceptibility. The ex-
panded view around 8 T, which is outside the CDS, shows
that the normalized signals are indeed identical. For higher
fields, on the other hand, the upper part of the susceptibili-
ty waveform, measured with the smaller modulation level,
is reduced. The expanded view around 13.5 T shows that
the signals are identical except for the positive part of the
oscillation. This implies that at this part of the dHvA oscil-
lation the magnetization is irreversible and there is a small
hysteresis loop. The width of the hysteresis loop is of the
order of 0.2 G.
A similar decrease of the normalized response was ob-
served earlier [17] on silver at low temperatures. Unfortu-
nately, because of the absence of the experimental para-
meters, this study can be compared only qualitatively with
our data.
The magnetic field where the normalized pickup vol-
tages start to differ between low and high modulation level
is marked approximately by an arrow in Fig. 3,a. We ob-
tain for the critical magnetic field 0 1 =cHμ 10.0 T.
Figure 3,b shows the behavior of the third harmonic
which was simultaneously measured with the first harmon-
ic response in Fig. 3,a for 0.2 G modulation amplitude. For
magnetic fields lower than the critical field there is only
noise. At the transition to the CDS hysteresis arises and the
third harmonic increases very steeply. This is nicely seen
in the respective expanded views. The critical field of the
CDS phase boundary can be obtained as the intersection of
the two straight lines shown in Fig. 3,b. Here, the critical
field is found as 0 2 = 9.6cHμ T.
The amplitude of the third harmonic is expected to go
to zero in each diamagnetic part of the dHvA period be-
cause the sample magnetization is homogeneous here and
without hysteresis. The presented behavior in Fig. 3,b does
not go to zero exactly which is certainly the result of a
small rectification effect or, what is the same, the result of
phase smearing of the oscillation signal. The homogeneity
of the coil is about 30 ppm in a sphere with 1 cm diameter
which may result in a field inhomogeneity of about 1 G in
the sample volume at 10 T. Therefore, the transition to the
CDS does not occur simultaneously in the whole sample.
This effect will be much bigger in a resistive coil where the
homogeneity is 20 times worse. However we will see be-
low that the third harmonic rectification does not affect the
determination of the critical field of the CDS.
In Fig. 4,a the phase angle of the third harmonic is
shown for the same conditions like in Fig. 3. For magnetic
Fig. 3. (Color online) (a) Pickup voltage normalized on the mod-
ulation level for low and high modulation level. Up to about 10 T
the response is linear with respect to the modulation level.
(b) Third harmonic of the pickup voltage measured at 0.2 G mod-
ulation amplitude showing that starting from 9.5 T the harmonic
content in the response increases steeply. Lower part of the figure
shows respective zooms. Both data measured at 2.7 K.
20
0
–20
0.4
0.2
0
6 8 10 12 14
�0H, T
3
h
ar
m
.
rd
V
p
,
�
V
n
o
rm
.
V
V
p
,
�
a
b
1.0 G 0.2 G
1.0 Ga
b
0.2 G
2
0
–2
0.4
0.2
0
8.002 8.004 13.480 13.484
�0H, T �0H, T
20
0
–20
0.4
0.2
0
10
–10
3
h
ar
m
.
rd
V
p
,
�
V
n
o
rm
.
V
V
p
,
�
3
h
ar
m
.
rd
V
p
,
�
V
n
o
rm
.
V
V
p
,
�
R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, and W. Joss
54 Fizika Nizkikh Temperatur, 2011, v. 37, No. 1
fields where the amplitude of the third harmonic is below
the noise level its phase angle is not determined. Therefore,
the phase varies between –180 to +180 degree. With the
appearance of a third harmonic signal at the transition to
the CDS the phase becomes finite. This passage has a thre-
shold character as well. The arrow in Fig. 4,a shows the
position of the threshold which yields the critical field
0 3 = 9.7cHμ T.
The behavior of the out-of-phase part of the first har-
monic response, shown in Fig. 4,b, offers another possibili-
ty to determine the critical field. In the uniform state, with-
out domains, the imaginary part is small and varies
smoothly especially at low magnetic field due to the mag-
netoresistance and changing eddy currents. After the tran-
sition to the CDS the out-of-phase signal changes rapidly.
The transition point can be found as the intersection of two
lines, as it is shown in Fig. 4,b. Here, we obtain for the
critical field 0 4 = 9.4cHμ T.
A comparison of the values 1... 4c cH for the transition
field at 2.7 K shows that they are very close. We note that
the critical field of about 10 T agrees roughly with the
phase boundary found in the Hall probe experiments [8].
All above presented methods could, in principle, be used to
determine the phase boundary of the CDS. The first me-
thod (Fig. 3,a) requires at least two field sweeps. Mea-
surements of the out-of-phase part (Fig. 3,b) are not pre-
cise, due to the high conductivity of silver and the resulting
eddy currents. (The situation might be different in a less
conducting metal.) Therefore, for silver the third harmonic
measurements to determine the CDS phase diagram are
preferred. Moreover, we will see below that the obtained
values with the third harmonic for the phase boundary
( , )H T do not depend drastically on the frequency and
amplitude of the field modulation which offers the possi-
bility to measure in noisier conditions. The found scatter-
ing in the values of the transition fields obtained from the
different methods gives an uncertainty of about ±0.5 T in
the transition fields.
4. Phase diagram
Because of the increased noise level in the water-cooled
resistive magnets, we needed to increase the signal-to-
noise ratio by using rather high modulation frequencies
≈160 Hz and higher modulation amplitudes, 1.0 G and
more. In the following we check whether the modulation
frequency and amplitude can be varied without changing
the value of the critical field deduced from the third har-
monic response.
Modulation amplitudes of the order of the width of the
hysteresis loop are required to resolve the amplitude reduc-
tion in the normalized pickup voltage in Fig. 3,a. For the
third harmonic signal, as shown in Fig. 2, the nonlinear
features persist up to high modulation amplitudes. Figure 5
shows traces with 0.2 and 1.0 G modulation amplitude. For
1.0 G modulation amplitude there is a very small third
harmonic signal before the transition to the CDS takes
place. This small contribution to the third harmonic is due
to the nonlinearity of the dHvA effect itself [3]. Neverthe-
less, the position of the sharp increase remains unchanged.
Figure 6 shows that increasing the modulation ampli-
tude up to 10 G and varying the modulation frequency by a
factor four between 40 and 160 Hz does not change the
position of the critical field, either. The results presented
here were obtained in the resistive magnet. The measure-
ments were made at low temperatures in order to compare
them with data obtained in the superconducting magnet.
All results for the CDS transition points obtained in the
superconducting and the resistive magnets are presented in
Fig. 7. The critical fields for each temperature are found as
the field where the third harmonic response starts to arise
like in Fig. 6. One should note that near the flat maximum
Fig. 5. (Color online) Third harmonic response for two modula-
tion levels measured in a superconducting coil at a modulation
frequency of 160 Hz at 2.7 K. The same critical field is found
from the steep increase.
2
1
0
0.4
0.2
0
6 8 10 12 14
�0H, T
3
h
ar
m
.
rd
V
p
,
�
V
a
b
1.0 G
0.2 G
Fig. 6. Third harmonic response measured after averaging over
the oscillations in a resistive coil for different modulation ampli-
tudes (a) and modulation frequencies (b). The critical field values
obtained at a given temperature are independent on modulation
frequency and amplitude.
4
2
0 0
1
4 6 8 10 2 4 6
a b
3
h
ar
m
.
rd
V
p
,
�
V
2.5 G
3
h
ar
m
.
rd
V
p
,
�
VT = 2.0 K T = 1.4 K
5.0 G
10.0 G
40 Hz
81 Hz
161 Hz
�0H, T �0H, T
Condon domain phase diagram for silver
Fizika Nizkikh Temperatur, 2011, v. 37, No. 1 55
of the phase diagram T-sweep measurements would be
in principle better. The solid line in Fig. 7 is the CDS
boundary calculated for silver using the LK formula for the
susceptibility criterion = 1χ with a Dingle temperature
of = 0.2 KDT for our sample [13,18]. For comparison,
theoretical diagrams for = 0.1DT K and 0.8 K are shown
in Fig. 7.
A good agreement of the phase diagram predictions
based on the LK formula with our data can be seen for a
Dingle temperature = 0.2DT K. Data points obtained in
superconducting and resistive magnets overlap which sup-
ports that the different measurement conditions did not
affect the precise determination of the phase boundary.
5. Conclusion
Like previously observed in beryllium [11], we have
shown that hysteresis appears in silver in the Condon do-
main state. This substantiates that hysteresis is likely to
occur in all pure metals that exhibit Condon domains. The
hysteresis leads to with threshold character arising ex-
tremely nonlinear response to a modulation field. In partic-
ular, the third harmonic response to a modulation field
increases sharply upon entering into the Condon domain
state. This offered the possibility to determine easily the
CDS phase boundary with high accuracy.
The critical fields obtained from the third harmonic of
the pickup signal of the ac modulation technique, turned
out to be independent on changes of the modulation fre-
quency and amplitude. Due to this independence this me-
thod could be used with higher modulation frequencies in
pulse magnetic fields.
Very good agreement of the CDS phase diagram is found
with calculations of the dHvA signal based on the LK
theory. This agreement shows that the LK formula de-
scribes well the field dependent magnetization of the near-
ly spherical Fermi surface of silver. Furthermore, the
agreement demonstrates that the described method is cor-
rect for the determination of the CDS phase diagram.
Acknowledgment
We are grateful to I. Sheikin and V.P. Mineev for fruit-
ful discussions.
1. J.H. Condon, Phys. Rev. 145, 526 (1966).
2. A.B. Pippard, Proc. R. Soc. A272, 192 (1963).
3. D. Shoenberg, Magnetic Oscillations in Metals, Cambridge
University Press, Cambridge (1984).
4. A. Gordon, I.D. Vagner, and P Wyder, Adv. Phys. 52, 385
(2003).
5. J.H. Condon and R.E Walstedt, Phys. Rev. Lett. 21, 612
(1968).
6. G. Solt, C. Baines, V.S. Egorov, D. Herlach, E. Krasnope-
rov, and U. Zimmermann, Phys. Rev. Lett. 76, 2575 (1996).
7. G. Solt and V.S. Egorov, Physica B318, 231 (2002).
8. R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M.
Jansen, and W. Joss, Phys. Rev. Lett. 95, 267209 (2005).
9. G. Solt, Solid State Commun. 118, 231 (2001).
10. G. Solt, C. Baines, V.S. Egorov, D. Herlach, U. Zimmermann,
Phys. Rev. B59, 6834 (1999).
11. R.B.G. Kramer, V.S. Egorov, A.G.M. Jansen, and W. Joss,
Phys. Rev. Lett. 95, 187204 (2005).
12. T. Klein, A. Conde-Gallardo, J. Marcus, C. Escribe-Filippini,
P. Samuely, P. Szabo, and A.G.M. Jansen, Phys. Rev. B58,
12411 (1998).
13. R.B.G. Kramer, V.S. Egorov, A. Gordon, N. Logoboy, W.
Joss, and V.A. Gasparov, Physica B362, 50 (2005).
14. V.A. Gasparov, Sov. Phys. JETP 41, 1129 (1975) [Zh. Eksp.
Teor. Fiz. 68, 2259 (1975)].
15. V.A. Gasparov and R. Huguenin, Adv. Phys. 42, 393 (1993).
16. V.B. Zernov and Y.V. Sharvin, Sov. Phys. JETP 9, 737
(1959) [Zh. Eksp. Teor. Fiz. 36, 1038 (1959)].
17. J.L. Smith and J.C. Lashley, J. Low Temp. Phys. 135, 161
(2004).
18. A. Gordon, M.A. Itskovsky, and P. Wyder, Phys. Rev. B59,
10864 (1999).
Fig. 7. (Color online) Phase diagram in the (H,T) plane for silver.
Experimental points from the superconducting and resistive mag-
net. The solid line is the CDS boundary calculated by the LK
formula for = 0.2DT K [13,18]. The dashed and the dotted lines
correspond to = 0.8DT and 0.1 K, respectively.
0
10 20 30 40 50
�0H, T
5
4
3
2
1
T
,
K
TD = 0.1 K
TD = 0.2 K
TD = 0.8 K
resistive magnet
superconducting magnet
|